This paper investigates the computational limits of modern Hopfield models through a fine-grained complexity analysis. The authors characterize a phase transition behavior in the efficiency of all possible modern Hopfield models based on the norm of patterns. They establish an upper bound criterion for the norm of input query patterns and memory patterns, showing that only below this criterion do sub-quadratic (efficient) variants of the modern Hopfield model exist, assuming the Strong Exponential Time Hypothesis (SETH). The paper provides a formal example of efficient constructions of modern Hopfield models using low-rank approximation when the efficient criterion holds. This includes a derivation of a lower bound on the computational time, scaling linearly with the maximum of the number of stored memory patterns and the length of input query sequences. The authors also prove the memory retrieval error bound and exponential memory capacity of the model. The paper presents a reduction from the Approximate Nearest Neighbor Search (ANNS) problem to the modern Hopfield model, showing that solving the ANNS problem requires sub-quadratic time under SETH. This implies that solving the modern Hopfield model also requires sub-quadratic time. The authors then present an efficient algorithm for the modern Hopfield model based on low-rank approximation, achieving almost linear time complexity. This algorithm maintains the exponential memory capacity characteristic of modern Hopfield models. The paper concludes with a discussion of the implications of these findings for the development of efficient Hopfield-based and transformer-based foundation models.This paper investigates the computational limits of modern Hopfield models through a fine-grained complexity analysis. The authors characterize a phase transition behavior in the efficiency of all possible modern Hopfield models based on the norm of patterns. They establish an upper bound criterion for the norm of input query patterns and memory patterns, showing that only below this criterion do sub-quadratic (efficient) variants of the modern Hopfield model exist, assuming the Strong Exponential Time Hypothesis (SETH). The paper provides a formal example of efficient constructions of modern Hopfield models using low-rank approximation when the efficient criterion holds. This includes a derivation of a lower bound on the computational time, scaling linearly with the maximum of the number of stored memory patterns and the length of input query sequences. The authors also prove the memory retrieval error bound and exponential memory capacity of the model. The paper presents a reduction from the Approximate Nearest Neighbor Search (ANNS) problem to the modern Hopfield model, showing that solving the ANNS problem requires sub-quadratic time under SETH. This implies that solving the modern Hopfield model also requires sub-quadratic time. The authors then present an efficient algorithm for the modern Hopfield model based on low-rank approximation, achieving almost linear time complexity. This algorithm maintains the exponential memory capacity characteristic of modern Hopfield models. The paper concludes with a discussion of the implications of these findings for the development of efficient Hopfield-based and transformer-based foundation models.